A circle with radius $4$ has a sector with a $270^\circ$ central angle. What is the area of the sector? ${16\pi}$ $\color{#9D38BD}{270^\circ}$ ${12\pi}$ ${4}$
Solution: First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. $A_c = \pi r^2$ $A_c = \pi (4)^2$ $A_c = 16\pi$ The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{270^\circ}{360^\circ} = \dfrac{A_s}{16\pi}$ $\dfrac{3}{4} = \dfrac{A_s}{16\pi}$ $\dfrac{3}{4} \times 16\pi = A_s$ $12\pi = A_s$